In a previous blog I highlighted the increasing importance of the choice of model for calculating the IE01 of an LPI liability, and hence the IE01 of a pension scheme. LPI is the term commonly used for a liability that is linked to RPI but subject to caps and floors on the annual changes in RPI. From now on I will refer to this as L-RPI for clarity. The IE01 of a pension scheme is the sensitivity of the liabilities to a one basis point change in inflation. The conclusion of the blog was that the choice of model could have material impact on the hedging portfolio that best matches the liabilities.

Since then, we have encountered another situation where this model choice becomes important: many schemes have CPI-linked liabilities and, in some cases, these liabilities also have caps and floors in place. Let us call these L-CPI liabilities. These introduce several degrees of complexity compared with the L-RPI liabilities we are used to dealing with:

** 1. Delta**

In the same way that the IE01 for an L-RPI liability becomes model dependent, this is also the case for an L-CPI liability. However to the extent that the actual level of CPI is below that of RPI, the percentage delta for an L-CPI[0,3] liability will be higher than the corresponding percentage delta for an L-RPI[0,3] liability, as illustrated in figure 1.

*Figure 1: Model IE01 of L-RPI[0,3] and L-CPI[0,3] swaps, expressed as a percentage of the IE01 of the corresponding RPI or CPI swap**Source: Underlying LPI curves, RBS; Calculations, Redington*

**2. Valuation**

The valuation point is more fundamental. With L-RPI liabilities, we can reference the swap curves produced by banks for year-on-year RPI cashflows with and without caps and floors. Even though it might be argued that these instruments trade relatively infrequently, it is still true that we can get close to tradable prices from a number of sources.

In the case of L-CPI this information is not available; the best we can do is

**the market level from what we know about how L-RPI trades relative to RPI.**

*imply*

*How can we do this?*The best we can do to derive a market consistent assumption for an L-CPI liability is to use what information we have about how L-RPI trades relative to RPI.

*This involves calibrating an inflation volatility model to the market quotes for L-RPI*. A model to use would be the SABR model, as implemented with year-on-year parameters. The SABR model is a stochastic volatility model which allows for the volatility skew in option prices, and is becoming market standard among banks and fund managers for this purpose.

Once we have this volatility model, we can then make a choice about how the different levels of CPI map to levels in RPI. This is a whole topic in itself, particularly in light of recent changes; however, for the purposes of this article let us assume that a fixed difference between RPI and CPI can be agreed upon, and let us say this is 50bps. With this assumption in hand we can then derive the implied zero-coupon curve for CPI; and further to that, we can then also calculate using our SABR model which was calibrated by reference to L-RPI quotes, the value of the caps and floors on the CPI index.

One complication is there are several different approaches one can take to “mapping” the inflation model assumptions as they apply to RPI, to the CPI case, and these can yield quite different results.

*This illustrates a fairly material model dependency in this calculation.*

What we find is that, for an L-CPI[0,3] liability, the different modelling assumptions generate a significant spread in the possible pricing basis compared to CPI. Our calculations suggest that L-CPI[0,3] could realistically price anywhere between a c30bps discount to a 10bps premium to the CPI curve, at different terms, depending on the detail of the modelling assumptions as shown by figure 2. This pricing basis reflects the relative price of the 0% floor and the 3% cap.

*Figure 2: pricing basis of L-CPI[0,3] relative to CPI, two different calculation methods***)**

*(Detail on methods below**Source: Data, RBS; Calculations, Redington*

Of course, as more pension schemes move to CPI indexing we can expect that there will be more of a market for CPI swaps, and presumably L-CPI swaps subsequently, so we can then adopt the fully market-consistent method we currently employ for L-RPI.

The upshot for pension schemes is that their IE01 may be extremely difficult to estimate and model if the liabilities are linked to CPI with caps and floors. Not only that, the value placed on the liabilities themselves could be highly dependent on the model. Trustees should be asking their actuaries and advisors their approach to this problem, and to clarify what the impact on the choice of model has on the value of the liabilities of the scheme.

**Detailed description of Methods 1 and 2**

Method 1 takes the implied volatility of each strike in RPI space, for example 0%, 3% and 5% levels, and maps this to the corresponding level on CPI. So the 0% strike in CPI space will be given the same volatility as the 0% strike in RPI space.

Method 2 allows for whatever assumption is being used for the difference between RPI and CPI when doing this mapping. For example, say CPI is fixed as 50bps below RPI. This method says that the volatility for a 0% strike in CPI should be the same as the volatility of a 0.5% strike in RPI space (0% + 50bps).

Another approach which would also be justifiable would be to scale the volatilities down appropriately in line with the realised differences between CPI and RPI volatility. The output of this method is not shown.

*[Please note that all opinions expressed in this blog are the author’s own and do not constitute investment advice. Click here for full disclaimer]*